arXiv subjects

Patrick White

Publications and source records attributed to Patrick White.

3 recordsLinked to original sources

arXiv2026

The small Davenport constant of the Heisenberg group of order 125

The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$ open. We settle the first open case: $\mathsf{d}(H_{125})=12$. The lower bound is the explicit product-one-free sequence $x^4y^4v^4$. For the upper bound we record a product-one criterion that reduces the non-commutative problem to additive combinatorics over $\mathbb{F}_5^2$, and then reduce "every length-13 sequence has a product-one subsequence" to a single finite statement -- a spread bound on quotient multisets -- which we verify by an exhaustive, memory-flat search in C, its verdict independently reproduced by a second search with a different pruning strategy. Every auxiliary lemma is machine-checked. The argument is genuinely $p$-specific: we identify the exact step that fails for $p\ge7$ (a Chevalley-Warning shortcut whose forced block need not be wide), exhibit the obstructing multiset for $p=7$, and leave only $18\le\mathsf{d}(H_{343})\le24$. The techniques -- the Cauchy-Davenport theorem, Chevalley-Warning, and Olson's value of the Davenport constant of $C_p^2$ -- are standard; the contribution is their assembly against a new non-abelian target and the finite verification that closes it.

math.CO↗

arXiv2026

New bounds for double covers of the discrete box {0,1,2}^d

A proper sub-box of $A=\{0,1,2\}^d$ is a product $S_1\times\dots\times S_d$ with each $\varnothing\neq S_i\subsetneq\{0,1,2\}$. A double cover is a finite multiset of proper sub-boxes covering every point of $A$ exactly twice; write $f(d)$ for the minimum size of a double cover. Leader, Milicevic and Tan asked whether $f(d)\ge 2^d$ for all $d$ (Question 4.1 of the PatternBoost paper of Charton-Ellenberg-Wagner-Williamson), analogous to the Alon-Bohman-Holzman-Kleitman partition bound $2^d$. No better than the trivial volume bound was previously known, for any $d\ge 2$. We prove the first nontrivial lower bounds. A modular refinement of the parity argument gives $f(d)\ge 2^{d+1}/(d+1)$; a slicing argument gives $f(4)\ge 19$, $f(5)\ge 33$, both above $2^d$, resolving the question for $d=4,5$ -- the first cases beyond the trivially known $d\le 3$. A finer "line rigidity" argument yields $f(6)\ge 60$, breaking the profile-statistic barrier (capped at $57$, shown here). This is formally verified in Lean 4: $f(6)\ge 60$ is machine-checked on the three standard Mathlib axioms alone. On the upper-bound side, a dimension-lifting construction $f(r+3)\le 6\cdot 2^r+3f(r)$ gives $f(6)\le 81$ (improving the known $82$) and $f(d)\le(\tfrac65+o(1))2^d$ asymptotically; a refinement improves the constant to $\tfrac87$. This makes partial progress on PatternBoost's problem of reducing their constant $1.28$, and refutes the closed-form guess $f(d)=5\cdot 2^{d-2}+1$ from $d=7$ on. Together, $60\le f(6)\le 81$. Finally we isolate the construction-side obstruction -- an "S+c=2^j+1" phenomenon, every skeleton sitting exactly one box past the partition bound -- and show it is of a piece with the Leader-Milicevic-Tan question itself.

math.CO↗

arXiv2010

Testing Closeness of Discrete Distributions

Given samples from two distributions over an $n$-element set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in $n$, specifically, $O(n^{2/3}\epsilon^{-8/3}\log n)$, independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than $\max\{\epsilon^{4/3}n^{-1/3}/32, \epsilon n^{-1/2}/4\}$) or large (more than $\epsilon$) in $\ell_1$ distance. This result can be compared to the lower bound of $\Omega(n^{2/3}\epsilon^{-2/3})$ for this problem given by Valiant. Our algorithm has applications to the problem of testing whether a given Markov process is rapidly mixing. We present sublinear for several variants of this problem as well.

cs.DS↗